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\title{Centralizer properties of \\
the affine Hecke algebra of type
C}
\author[Zajj Daugherty]{Zajj Daugherty\\
(joint with Arun Ram)}
\institute{Dartmouth College}
\date{January 11, 2013}
\begin{document}



\begin{frame}{}\vfill
\centerline{\begin{tikzpicture}[scale=2]
\filldraw[structure,rounded corners=8pt] (0,0)--(0,1)--(5,1)--(5,0);
\filldraw[structure,rounded corners=8pt] (0,-2)--(0,-2.5)--(5,-2.5)--(5,-2);
\draw[thick, structure] (0,0)--(0,-2) (5,0)--(5,-2);
\node[white, font=\bfseries] at (2.5,.6) {\Huge {H~E~L~L~O}};
\node[white] at (2.5,.2) {\large my name is };
\node[font=\bfseries] at (2.5,-.6) {\Huge Zajj Daugherty};
\node at (2.5,-1.1) {\large John Wesley Young Research Instructor};
\node at (2.5,-1.35) {\large Dartmouth College};
\node at (2.5,-1.75) {\large Postdoctoral Researcher, ICERM};
\end{tikzpicture}}\vfill
\end{frame}

\begin{frame}{\bf A little about me}
\vfill
\begin{itemize}
	\item \alert{\large \bf Harvey Mudd College}, BS 2005\\
	Thesis: ``An algebraic approach to Voting Theory'' 
	\vfill
	\item \alert{\large \bf University of WI -- Madison}, Ph.D 2010\\
	Advisor: Arun Ram\\
	Thesis: ``Two-boundary centralizer algebras''
	\vfill
	\item \alert{\large \bf St. Olaf College}, Visiting Assistant Professor 2010-2011
	\vfill
	\item \alert{\large \bf Dartmouth College}, JWY Research Instructor 2011-2014
\end{itemize}
\vfill
More at \,\texttt{http://www.math.dartmouth.edu/$\sim$zdaugherty/}
\vfill
\end{frame}





\begin{frame}{\bf A few of my favorite things: Schur-Weyl duality }
{} \small \alert{\bf Classical example:} (Schur 1901) \\\pause
\vspace{-.1in}
\begin{enumerate}[\qquad]
\item  $\GL_n(\CC)$ acts on $\CC^n \otimes \CC^n \otimes \cdots \otimes \CC^n = \left( \CC^n\right)^{\otimes k}$
diagonally.
$$g \cdot (v_1 \otimes v_2 \otimes \cdots \otimes v_k) = g v_1 \otimes g v_2 \otimes \cdots \otimes g v_k.$$
\item \pause $S_k$ also acts on  $\left( \CC^n\right)^{\otimes k}$ by place permutation.
$$\begin{tikzpicture}[line width=1pt, scale=.75]
\foreach \x in {1,...,5}{
 \filldraw [black] (\x,0) circle (2pt);
 \filldraw [black] (\x,.75) circle (2pt);
\node at (\x, -.3) {\small $v_{\x}$};
}
\foreach \x in {1,...,4}{
\node at (\x+.5, -.3) {\small $\otimes$};
\node at (\x+.5, 1) {\small $\otimes$};
}
\node at (1, 1) {\small $v_{3}$};
\node at (2, 1) {\small $v_{1}$};
\node at (3, 1) {\small $v_{5}$};
\node at (4, 1) {\small $v_{2}$};
\node at (5, 1) {\small $v_{4}$};
\draw[->, shorten >=2pt] (1,0)--(2,.75); 
\draw[->, shorten >=2pt] (2,0)--(4,.75); 
\draw[->, shorten >=2pt] (3,0)--(1,.75); 
\draw[->, shorten >=2pt] (4,0)--(5,.75); 
\draw[->, shorten >=2pt] (5,0)--(3,.75); 
\end{tikzpicture}$$
\end{enumerate}
\vspace{-.1in}

\only<4-5>{
$$\begin{tikzpicture}[line width=1pt, scale=.75]
\uncover<4->{
	\foreach \x in {1,...,5}{
	\draw[dotted, structure] (\x,0)--(\x,-.5);
	}
}
%%%%%%%%%%%%%
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 \filldraw [black] (\x,1.5) circle (2pt);
 \filldraw [black] (\x,0) circle (2pt);
}
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\draw[<-, shorten <=2pt] (4,1.5)--(5,0); 
\draw[<-, shorten <=2pt] (5,1.5)--(3,0); 
%%%%%%%%%%%%%
\pgftransformyshift{-2.1cm}
%%%%%%%%%%%%%
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	\draw[<-, shorten <=2pt] (5,1.5)--(3,0); 
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%%%%%%%%%%%%%
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%%%%%%%%%%%%%
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	\draw[<-, shorten <=2pt] (4,1.5)--(3,0); 
	\draw[<-, shorten <=2pt] (5,1.5)--(4,0); 
}
\end{tikzpicture}$$
{(multiplication given by concatenation)}
}
\uncover<6->{
These actions commute! Moreover, they centralize each other. \\}
\uncover<7->{\alert{\bf Why this is exciting:}\\
{\small In general, if \\
		$\qquad V$ is a vector space,\\
		$\qquad \cA \subseteq \End(V)$, and \\
		$\qquad \cB = \End_{\cA}(V) = \{b \in \End(V) ~|~ ab = ba \}$.\\
		Then under `good' conditions,
		commuting actions provides a duality between the $\cA$-modules and the $\cB$-modules inside of $V$.}}
\end{frame}

\begin{frame}{\bf A few of my favorite things: Diagram algebras}
Start with an algebra $\cA$ (e.g.\\
\begin{enumerate}[~~]
\item $\cA$ is a group algebra, like $\CC S_n$, or \\
\item $\cA = U\fg$ is the enveloping algebra of a favorite Lie algebra, or 
\item $\cA = U_q\fg$ is the quantum group associated to a fav Lie alg, etc..)
\end{enumerate}\pause
Pick some $\cA$-modules, and build a tensor space.  
\\
What operators commute with the action of $\cA?$ 
\pause

~

\alert{\large Example:}
Let $\cA = U\fgl_n$,  $M= \CC^n$, and $V = M^{\otimes k}$ $(n>k)$.\\
Then commuting operators can be encoded as permutation diagrams: 

\vspace{-.1 in}


$$\begin{tikzpicture}[line width=1pt]
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  \node[above] at (\x, 1) {\small $M$};}
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\node[below] at (\x+.5, 0) {\small $\otimes$};
\node[above] at (\x+.5, 1) {\small $\otimes$};
}
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\node at (2, 1.3) {\small $v_{1}$};
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\draw[shorten >=2pt] (5,0)--(3,1); }\only<1-3>{
\draw[->, shorten >=2pt] (1,0)--(2,1); 
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\draw[->, shorten >=2pt] (5,0)--(3,1); }
\uncover<4->{\node[right] at (5.5,.5) {\LARGE $\uparrow$ \normalsize $b$};}
\end{tikzpicture}$$

\vspace{-.1 in}

\pause {\scriptsize (Take the algebra generated linearly, with multiplication given by concatenation)}

\end{frame}

\begin{frame}{\bf A few of my favorite things: Diagram algebras}
{Recall: we're starting with an algebra $\cA$ with module $V$,\\
 and looking for an algebra $\cB$ so that 
 $ab = ba$ in $\End (V)$. } 


{\bf More examples:}

\only<1>{	\begin{enumerate}[\qquad]
		\item $\cA = U\fsl_2$, and 
		\item $M=\CC^2$ is the defining representation and $V = M^{\otimes k}$.
		\item Then $\cB$ is the \emph{Temperley-Lieb algebra}: 
	\end{enumerate}
$$\begin{tikzpicture}[line width=1pt]
\foreach \x in {1,...,5}{
 \filldraw [black] (\x,1.5) circle (2pt);
 \filldraw [black] (\x,0) circle (2pt);
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  \node[above] at (\x, \TOP) {\small $M$};
}
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\node[above] at (\x+.5, \TOP) {\small $\otimes$};
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 \HzBot[3,4];
\draw (3,1.5)--(1,0); 
\draw (4,1.5)--(2,0); 
\draw (5,1.5)--(5,0);
\node at (3,-.75) {\scriptsize\alert{non-crossing pairings}}; 
\end{tikzpicture}$$
}\only<2>{	\begin{enumerate}[\qquad]
		\item $\cA = U\fg$ where $\fg = \fso_n$ or $\fsp_{2n}$, and 
		\item $M$ is the defining representation and $V = M^{\otimes k}$.
		\item Then $\cB$ is the \emph{Brauer algebra}: 
	\end{enumerate}
$$\begin{tikzpicture}[line width=1pt]
\foreach \x in {1,...,5}{
 \filldraw [black] (\x,1.5) circle (2pt);
 \filldraw [black] (\x,0) circle (2pt);
  \node[below] at (\x, 0) {\small $M$};
  \node[above] at (\x, \TOP) {\small $M$};
}
\foreach \x in {1,...,4}{
\node[below] at (\x+.5, 0) {\small $\otimes$};
\node[above] at (\x+.5, \TOP) {\small $\otimes$};
}
\HzTop[1,2];
 \HzBot[2,4];
\draw (3,1.5)--(1,0); 
\draw (4,1.5)--(5,0); 
\draw (5,1.5)--(3,0); 
\node at (3,-.75) {\scriptsize\alert{possibly crossing pairings}}; 
\end{tikzpicture}$$}\only<3>{	\begin{enumerate}[\qquad]
		\item $\cA = U_q\fg$ (quantum group) where $\fg = \fso_n$ or $\fsp_{2n}$, and 
		\item $M$ is the defining representation and $V = M^{\otimes k}$.
		\item Then $\cB$ is the \emph{Birman-Murakami-Wenzl (BMW) algebra}: 
	\end{enumerate}
$$\begin{tikzpicture}[line width=1pt]
\HzTop[1,2];
 \HzBot[2,4];
\draw (3,1.5)--(1,0); 
\draw[over] (4,1.5)--(5,0); 
\draw[over] (5,1.5)--(3,0); 
\foreach \x in {1,...,5}{
 \filldraw [black] (\x,1.5) circle (2pt);
 \filldraw [black] (\x,0) circle (2pt);
  \node[below] at (\x, 0) {\small $M$};
  \node[above] at (\x, \TOP) {\small $M$};
}
\foreach \x in {1,...,4}{
\node[below] at (\x+.5, 0) {\small $\otimes$};
\node[above] at (\x+.5, \TOP) {\small $\otimes$};
}
\node at (3,-.75) {\scriptsize\alert{tangles}}; 
\end{tikzpicture}$$}\only<4>{	\begin{enumerate}[\qquad]
		\item $\cA = U_q\fg$ (quantum group) where $\fg$ can be many things and 
		\item $L$ and $M$ are $\cA$-modules, and  $V = M\otimes L^{\otimes k}$.
		\item Then $\cB$ is a quotient of the grp alg of the \emph{affine braid group}: 
	\end{enumerate}
$$\begin{tikzpicture}[line width=1pt]
\PoleTop[\TOP-.2]
\PoleBottom[.2]
	\draw (2,1.5).. controls (4,1) .. (5,0);
	\Under[1,1.5][1.75,.75];
	\Under[.25,.75][2,0];
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	\Under[4,1.5][4.25,.75];
	\Pole[.5*\TOP][\TOP]
	\Over[3,1.5][.25,.75];
	\Over[5,1.5][4,.75];
	\Over[1.75,.75][1,0];
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		 \filldraw [black] (\x,1.5) circle (2pt);
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\node[below] at (.55, 0) {\small $M \otimes$};
\node[above] at (.55, \TOP) {\small $M \otimes$};
\node at (3,-.75) {\scriptsize\alert{braids in a space with one puncture}}; 
\end{tikzpicture}$$}
\begin{center}{\scriptsize 
(Take the algebra generated linearly, \\
with multiplication given by concatenation\only<4>{)}\uncover<1-3>{,\\ 
and relations on closed loops)}\\
}
\end{center}
\end{frame}

\begin{frame}{\bf My pet examples}
\begin{enumerate}[]
		\item $\cA = U\fg$ or $U_q\fg$ where $\fg = \fso_n$ or $\fsp_{2n}$,
		\item $L$ is the defining representation and $V = M\otimes L^{\otimes k}$.
	\end{enumerate}
$$\begin{tikzpicture}[line width=1pt, scale=.75]
\node at (3,\TOP+2) {\alert{Quantum group case:}};
\node at (3,\TOP+1.25) {\alert{Affine BMW algebra}};
\Under[.25,.75][1,0];
{\PoleTop[\TOP]
\PoleBottom[0]
\PoleO[0][\TOP]}
\HzTop[1,2];
 \HzBot[2,4];
	\Over[3,1.5][.25,.75];
\draw[over] (4,1.5)--(5,0); 
\draw[over] (5,1.5)--(3,0); 
\foreach \x in {1,...,5}{
 \filldraw [black] (\x,1.5) circle (2pt);
 \filldraw [black] (\x, 0) circle (2pt);
 \node at (\x, 2) {\scriptsize $L$};
  \node at (\x, -.5) {\scriptsize $L$};
}
\foreach \x in {1,...,4}{
 \node at (\x+.5, 2) {\scriptsize $\otimes$};
  \node at (\x+.5, -.5) {\scriptsize $\otimes$};
}
{ \node at (.415, 2) {\scriptsize $M$};
  \node at (.415, -.5) {\scriptsize $M$};
 \node at (.725, 2) {\scriptsize $\otimes$};
  \node at (.725, -.5) {\scriptsize $\otimes$};}
\node at (3,-1) {\scriptsize\alert{tangles in a space with one puncture}}; 
\end{tikzpicture}
\qquad
\begin{tikzpicture}[line width=1pt, scale=.75]
\node at (3,\TOP+2) {\alert{Enveloping algebra case:}};
\node at (3,\TOP+1.25) {\alert{Degenerate affine BMW algebra}};
\HzTop[1,2];
 \HzBot[2,4];
\draw (3,1.5)--(1,0); 
\draw (4,1.5)--(5,0); 
\draw (5,1.5)--(3,0); 
\foreach \x in {1,...,5}{
 \filldraw [black] (\x,1.5) circle (2pt);
 \filldraw [black] (\x, 0) circle (2pt);
 \node at (\x, 2) {\scriptsize $L$};
  \node at (\x, -.5) {\scriptsize $L$};
}
\foreach \x in {1,...,4}{
 \node at (\x+.5, 2) {\scriptsize $\otimes$};
  \node at (\x+.5, -.5) {\scriptsize $\otimes$};
}
{ \node at (.415, 2) {\scriptsize $M$};
  \node at (.415, -.5) {\scriptsize $M$};
 \node at (.725, 2) {\scriptsize $\otimes$};
  \node at (.725, -.5) {\scriptsize $\otimes$};}
\filldraw [structure] (1.55,.4) circle (4pt);
\node at (3,-1) {\scriptsize\alert{decorated pairings}}; 
\end{tikzpicture}
$$
Papers:
\begin{enumerate}
\item \alert{Affine and degenerate affine BMW algebras: The center} (with Arun Ram and Rahbar Virk), to appear in Osaka Journal of Mathematics. arXiv:1105.4207
\item \alert{Affine and degenerate affine BMW algebras: Actions on tensor space} (with Arun Ram and Rahbar Virk), to appear in Selecta Mathematica. arXiv:1205.1852
\end{enumerate}

\end{frame}


\begin{frame}{\bf My pet examples}
\begin{enumerate}[]
		\item $\cA = U\fg$ or $U_q\fg$ where $\fg$ can be many things,
		\item $M$, $N$, and $L$ are $\cA$-modules and $V = M\otimes L^{\otimes k} \otimes N$.
	\end{enumerate}
	\def\K{5}
$$\begin{tikzpicture}[line width=1pt]
\node at (3,2.6) {\alert{The two-boundary braid group}};
\node at (3,2.2) {\alert{and its degenerate version}};
\SymPolesTop[\TOP-.2][\K]
\SymPolesBottom[.2][\K]
	\draw (2,1.5)-- (5,0);
	\Over[1,1.5][1,0];
	\PoleO[0][.5*\TOP]
	\Pole[.5*\TOP][\TOP]
	\Over[3,1.5][.25,.75];
	\RPoleO[0,\TOP]
\draw[over] (4,\TOP)--(3,0); 
\Under[5,\TOP][5.85,.5*\TOP] 
\RRPoleO[0,\TOP][\K]
	\Over[.25,.75][2,0];
	\Over[5.85,.5*\TOP][4,0];
	\foreach \x in {1,...,5}{
		 \filldraw [black] (\x,1.5) circle (2pt);
		 \filldraw [black] (\x,0) circle (2pt);
		  \node[below] at (\x, 0) {\small $L$};
		  \node[above] at (\x, \TOP) {\small $L$};
	 }
\foreach \x in {1,...,4}{
\node[below] at (\x+.5, 0) {\small $\otimes$};
\node[above] at (\x+.5, \TOP) {\small $\otimes$};
}
\node[below] at (.55, 0) {\small $M \otimes$};
\node[above] at (.55, \TOP) {\small $M \otimes$};
\node[below] at (5.45, 0) {\small $\otimes N$};
\node[above] at (5.45, \TOP) {\small $\otimes N$};
\node at (3,-.75) {\scriptsize\alert{braids in a space with two punctures}}; 
\end{tikzpicture}
$$



Papers:
\begin{enumerate}
\item \alert{Degenerate two-boundary centralizer algebras}, Pac.\ J.\ of Math., 258-1 (2012) 91--142. arXiv:1007.3950; see also doctoral dissertation, UW--Madison, June 2010.
\item In progress: \alert{Two boundary Hecke Algebras and \\the combinatorics of type $(C_n^{\vee}, C)$ Hecke algebras}\\ (with Arun Ram) 
\end{enumerate}

\end{frame}

\begin{frame}{\bf My pet examples}

\begin{enumerate}[]
		\item $\cA = \CC S_n$,
		\item $M$ is the permutation module or its large irreducible submodule.
	\end{enumerate}
$$\begin{tikzpicture}[line width=1pt, scale=.75]
\node at (3,\TOP+2) {\alert{Permutation module:}};
\node at (3,\TOP+1.25) {\alert{Partition algebra}};
\node at (3,-1.25) {\scriptsize \alert{connected components}};
\foreach \x in {1,...,5}{
 \filldraw [black] (\x,1.5) circle (2pt);
 \filldraw [black] (\x, 0) circle (2pt);
 \node at (\x, 2) {\scriptsize $M$};
  \node at (\x, -.5) {\scriptsize $M$};
}
\foreach \x in {1,...,4}{
 \node at (\x+.5, 2) {\scriptsize $\otimes$};
  \node at (\x+.5, -.5) {\scriptsize $\otimes$};
}
\draw (1,0)--(2,0)--(1,\TOP)--(1,0) (3,\TOP)--(4,0)--(5,0)--(3,\TOP);
\HzTop[2,4];
\end{tikzpicture}
\qquad 
\begin{tikzpicture}[line width=1pt, scale=.75]
\node at (3,\TOP+2) {\alert{Irreducible submodule:}};
\node at (3,\TOP+1.25) {\alert{Quasi-partition algebra}};
\node at (3,-1.25) {\scriptsize \alert{c.c\ w/ no isolated vertices}};
\foreach \x in {1,...,5}{
 \filldraw [black] (\x,1.5) circle (2pt);
 \filldraw [black] (\x, 0) circle (2pt);
 \node at (\x, 2) {\scriptsize $M$};
  \node at (\x, -.5) {\scriptsize $M$};
}
\foreach \x in {1,...,4}{
 \node at (\x+.5, 2) {\scriptsize $\otimes$};
  \node at (\x+.5, -.5) {\scriptsize $\otimes$};
}
\draw (1,0)--(2,0)--(1,\TOP)--(1,0) (3,\TOP)--(5,0)--(3,0)--(3,\TOP) (4,\TOP)--(5,\TOP);
\HzTop[2,4];
\end{tikzpicture}
$$

Paper:\\
 \alert{Quasi-partition algebra} (with Rosa Orellana), arXiv:1212.2596
\end{frame}

\end{document}